A retail store has an annual demand for 30,000 units of one of its products ordered from a wholesaler at a unit cost of $25. The cost of ordering has been estimated to be $10 per order. The company is fortunate in that if the item is not on hand when demanded the customer will wait until the next shipment. Administrative costs of backordering average $6 per unit per week. In addition to this, the retailer reduces the price of the product by $1 per unit for every week that delivery is delayed in order to keep the customer waiting for the order. Storage costs amount to $0.05 per unit per day. Assume a year has exactly 52 weeks or 364 days. Find:

a) the optimal order quantity

b) the optimal backorder quantity

c) the maximum number of units that will be in stock

d) the length of the inventory cycle

e) the fraction of time a customer has to wait to have his order filled.

Please show detailed solution using optimal order quantity derivation